6.04d Integration: for centre of mass of laminas/solids

Answer only one of the following two alternatives. EITHER \includegraphics{figure_11a} The diagram shows a uniform thin rod \(AB\) of length \(3a\) and mass \(8m\). The end \(A\) is rigidly attached to the surface of a sphere with centre \(O\) and radius \(a\). The rod is perpendicular to the surface of the sphere. The sphere consists of two parts: an inner uniform solid sphere of mass \(m\) and radius \(a\) surrounded by a thin uniform spherical shell of mass \(m\) and also of radius \(a\). The horizontal axis \(l\) is perpendicular to the rod and passes through the point \(C\) on the rod where \(AC = a\).

1. Show that the moment of inertia of the object, consisting of rod, shell and inner sphere, about the axis \(l\) is \(\frac{289}{15}ma^2\). [6]

The object is free to rotate about the axis \(l\). The object is held so that \(CA\) makes an angle \(\alpha\) with the downward vertical and is released from rest.

1. Given that \(\cos \alpha = \frac{1}{6}\), find the greatest speed achieved by the centre of the sphere in the subsequent motion. [6]

OR The times taken to run \(200\) metres at the beginning of the year and at the end of the year are recorded for each member of a large athletics club. The time taken, in seconds, at the beginning of the year is denoted by \(x\) and the time taken, in seconds, at the end of the year is denoted by \(y\). For a random sample of \(8\) members, the results are shown in the following table.

Member	A	B	C	D	E	F	G	H
\(x\)	24.2	23.8	22.8	25.1	24.5	24.0	23.8	22.8
\(y\)	23.9	23.6	22.8	24.5	24.2	23.5	23.6	22.7

\([\Sigma x = 191, \quad \Sigma x^2 = 4564.46, \quad \Sigma y = 188.8, \quad \Sigma y^2 = 4458.4, \quad \Sigma xy = 4510.99.]\)

1. Find, showing all necessary working, the equation of the regression line of \(y\) on \(x\). [4]

The athletics coach believes that, on average, the time taken by an athlete to run \(200\) metres decreases between the beginning and the end of the year by more than \(0.2\) seconds.

1. Stating suitable hypotheses and assuming a normal distribution, test the coach's belief at the \(10\%\) significance level. [8]

\includegraphics{figure_5} A thin uniform rod \(AB\) has mass \(kM\) and length \(2a\). The end \(A\) of the rod is rigidly attached to the surface of a uniform hollow sphere with centre \(O\), mass \(kM\) and radius \(2a\). The end \(B\) of the rod is rigidly attached to the circumference of a uniform ring with centre \(C\), mass \(M\) and radius \(a\). The points \(C\), \(B\), \(A\), \(O\) lie in a straight line. The horizontal axis \(L\) passes through the mid-point of the rod and is perpendicular to the rod and in the plane of the ring (see diagram). The object consisting of the rod, the ring and the hollow sphere can rotate freely about \(L\).

1. Show that the moment of inertia of the object about \(L\) is \(\frac{5}{2}(8k + 3)Ma^2\). [6] The object performs small oscillations about \(L\), with the ring above the sphere as shown in the diagram.
2. Find the set of possible values of \(k\) and the period of these oscillations in terms of \(k\). [6]

A uniform rod \(AB\), of length \(2a\) and mass \(2m\), can rotate freely in a vertical plane about a smooth horizontal axis through \(A\). A small rough ring of mass \(m\) is threaded on the rod. The rod is held in a horizontal position with the ring at rest at the mid-point of the rod. The rod is released from rest. Using energy considerations, show that, until the ring slides, $$a\dot{\theta}^2 = \frac{18}{11}g \sin \theta,$$ where \(\theta\) is the angle turned through by the rod. [3] Show that, until the ring slides, the magnitudes of the friction force and normal contact force acting on the ring are \(\frac{20}{11}mg \sin \theta\) and \(\frac{2}{11}mg \cos \theta\) respectively. [6] The coefficient of friction between the ring and the rod is \(\mu\). Find, in terms of \(\mu\), the value of \(\theta\) when the ring starts to slide. [2]

Answer only one of the following two alternatives. **EITHER** \includegraphics{figure_11a} A uniform plane object consists of three identical circular rings, \(X\), \(Y\) and \(Z\), enclosed in a larger circular ring \(W\). Each of the inner rings has mass \(m\) and radius \(r\). The outer ring has mass \(3m\) and radius \(R\). The centres of the inner rings lie at the vertices of an equilateral triangle of side \(2r\). The outer ring touches each of the inner rings and the rings are rigidly joined together. The fixed axis \(AB\) is the diameter of \(W\) that passes through the centre of \(X\) and the point of contact of \(Y\) and \(Z\) (see diagram). It is given that \(R = \left(1 + \frac{2}{3}\sqrt{3}\right)r\).

1. Show that the moment of inertia of the object about \(AB\) is \(\left(7 + 2\sqrt{3}\right)mr^2\). [8]

The line \(CD\) is the diameter of \(W\) that is perpendicular to \(AB\). A particle of mass \(9m\) is attached to \(D\). The object is now held with its plane horizontal. It is released from rest and rotates freely about the fixed horizontal axis \(AB\).

1. Find, in terms of \(g\) and \(r\), the angular speed of the object when it has rotated through \(60°\). [6]

**OR** Fish of a certain species live in two separate lakes, \(A\) and \(B\). A zoologist claims that the mean length of fish in \(A\) is greater than the mean length of fish in \(B\). To test his claim, he catches a random sample of 8 fish from \(A\) and a random sample of 6 fish from \(B\). The lengths of the 8 fish from \(A\), in appropriate units, are as follows. $$15.3 \quad 12.0 \quad 15.1 \quad 11.2 \quad 14.4 \quad 13.8 \quad 12.4 \quad 11.8$$ Assuming a normal distribution, find a 95% confidence interval for the mean length of fish in \(A\). [5] The lengths of the 6 fish from \(B\), in the same units, are as follows. $$15.0 \quad 10.7 \quad 13.6 \quad 12.4 \quad 11.6 \quad 12.6$$ Stating any assumptions that you make, test at the 5% significance level whether the mean length of fish in \(A\) is greater than the mean length of fish in \(B\). [7] Calculate a 95% confidence interval for the difference in the mean lengths of fish from \(A\) and from \(B\). [2]

Answer only one of the following two alternatives. EITHER \includegraphics{figure_10a} An object is formed by attaching a thin uniform rod \(PQ\) to a uniform rectangular lamina \(ABCD\). The lamina has mass \(m\), and \(AB = DC = 6a\), \(BC = AD = 3a\). The rod has mass \(M\) and length \(3a\). The end \(P\) of the rod is attached to the mid-point of \(AB\). The rod is perpendicular to \(AB\) and in the plane of the lamina (see diagram). Show that the moment of inertia of the object about a smooth horizontal axis \(l_1\), through \(Q\) and perpendicular to the plane of the lamina, is \(3(8m + M)a^2\). [4] Show that the moment of inertia of the object about a smooth horizontal axis \(l_2\), through the mid-point of \(PQ\) and perpendicular to the plane of the lamina, is \(\frac{3}{4}(17m + M)a^2\). [2] Find expressions for the periods of small oscillations of the object about the axes \(l_1\) and \(l_2\), and verify that these periods are equal when \(m = M\). [8] OR A farmer \(A\) grows two types of potato plants, Royal and Majestic. A random sample of 10 Royal plants is taken and the potatoes from each plant are weighed. The total mass of potatoes on a plant is \(x\) kg. The data are summarised as follows. $$\Sigma x = 42.0 \qquad \Sigma x^2 = 180.0$$ A random sample of 12 Majestic plants is taken. The total mass of potatoes on a plant is \(y\) kg. The data are summarised as follows. $$\Sigma y = 57.6 \qquad \Sigma y^2 = 281.5$$ Test, at the 5% significance level, whether the population mean mass of potatoes from Royal plants is the same as the population mean mass of potatoes from Majestic plants. You may assume that both distributions are normal and you should state any additional assumption that you make. [9] A neighbouring farmer \(B\) grows Crown potato plants. His plants produce 3.8 kg of potatoes per plant, on average. Farmer \(A\) claims that her Royal plants produce a higher mean mass of potatoes than Farmer \(B\)'s Crown plants. Test, at the 5% significance level, whether Farmer \(A\)'s claim is justified. [5]

\includegraphics{figure_3} A uniform disc, of radius \(a\) and mass \(2M\), is attached to a thin uniform rod \(AB\) of length \(6a\) and mass \(M\). The rod lies along a diameter of the disc, so that the centre of the disc is a distance \(x\) from \(A\) (see diagram).

1. Find the moment of inertia of the object, consisting of disc and rod, about a fixed horizontal axis \(l\) through \(A\) and perpendicular to the plane of the disc. [4]

The object is free to rotate about the axis \(l\). The object is held with \(AB\) horizontal and is released from rest. When \(AB\) makes an angle \(\theta\) with the vertical, where \(\cos \theta = \frac{3}{5}\), the angular speed of the object is \(\sqrt{\left(\frac{2g}{5a}\right)}\).

1. Find the possible values of \(x\). [5]

An object is formed from a uniform circular disc, of radius \(2a\) and mass \(3M\), and a uniform rod \(AB\), of length \(4a\) and mass \(kM\), where \(k\) is a constant. The centre of the disc is \(O\). The end \(B\) of the rod is rigidly joined to a point on the circumference of the disc so that \(OBA\) is a straight line. The fixed horizontal axis \(l\) is in the plane of the object, passes through \(A\) and is perpendicular to \(AB\).

1. Show that the moment of inertia of the object about the axis \(l\) is \(3Ma^2(26 + k)\). [5]
2. The object is free to rotate about \(l\). Show that small oscillations of the object about \(l\) are approximately simple harmonic. Given that the period of these oscillations is \(4\pi\sqrt{\frac{a}{g}}\), find the value of \(k\). [6]

\includegraphics{figure_5} A thin uniform rod \(AB\) has mass \(\lambda M\) and length \(2a\). The end \(A\) of the rod is rigidly attached to the surface of a uniform hollow sphere (spherical shell) with centre \(O\), mass \(3M\) and radius \(a\). The end \(B\) of the rod is rigidly attached to the surface of a uniform solid sphere with centre \(C\), mass \(5M\) and radius \(a\). The rod lies along the line joining the centres of the spheres, so that \(CBAO\) is a straight line. The horizontal axis \(L\) is perpendicular to the rod and passes through the point of the rod that is a distance \(\frac{1}{2}a\) from \(B\) (see diagram). The object consisting of the rod and the two spheres can rotate freely about \(L\).

1. Show that the moment of inertia of the object about \(L\) is \(\left(\frac{408 + 7\lambda}{12}\right)Ma^2\). [6]

The period of small oscillations of the object about \(L\) is \(5\pi\sqrt{\left(\frac{2a}{g}\right)}\).

1. Find the value of \(\lambda\). [6]

\includegraphics{figure_3} An object is made from a uniform solid hemisphere of radius \(0.56\) m and centre \(O\) by removing a hemisphere of radius \(0.28\) m and centre \(O\). The diagram shows a cross-section through \(O\) of the object.

1. Calculate the distance of the centre of mass of the object from \(O\). [4] [The volume of a hemisphere is \(\frac{2}{3}\pi r^3\).]
2. The object has weight \(24\) N. A uniform hemisphere \(H\) of radius \(0.28\) m is placed in the hollow part of the object to create a new uniform hemisphere with centre \(O\). The centre of mass of the non-uniform hemisphere is \(0.15\) m from \(O\). Calculate the weight of \(H\). [3]

\includegraphics{figure_2} An object is made from a uniform solid hemisphere of radius \(0.56\) m and centre \(O\) by removing a hemisphere of radius \(0.28\) m and centre \(O\). The diagram shows a cross-section through \(O\) of the object.

1. Calculate the distance of the centre of mass of the object from \(O\). [4] [The volume of a hemisphere is \(\frac{2}{3}\pi r^3\).]
2. Calculate the weight of \(H\). [3]

The object has weight \(24\) N. A uniform hemisphere \(H\) of radius \(0.28\) m is placed in the hollow part of the object to create a non-uniform hemisphere with centre \(O\). The centre of mass of the non-uniform hemisphere is \(0.15\) m from \(O\).

\includegraphics{figure_7} A uniform solid cone has height \(1.2 \text{ m}\) and base radius \(0.5 \text{ m}\). A uniform object is made by drilling a cylindrical hole of radius \(0.2 \text{ m}\) through the cone along the axis of symmetry (see diagram).

1. Show that the height of the object is \(0.72 \text{ m}\) and that the volume of the cone removed by the drilling is \(0.0352\pi \text{ m}^3\). [4]

[The volume of a cone is \(\frac{1}{3}\pi r^2 h\).]

1. Find the distance of the centre of mass of the object from its base. [6]

A cylindrical container is open at the top. The curved surface and the circular base of the container are both made from the same thin uniform material. The container has radius \(0.2 \text{ m}\) and height \(0.9 \text{ m}\).

1. Show that the centre of mass of the container is \(0.405 \text{ m}\) from the base. [3]

The container is placed with its base on a rough inclined plane. The container is in equilibrium on the point of slipping down the plane and also on the point of toppling.

1. Find the coefficient of friction between the container and the plane. [3]

\includegraphics{figure_4} The diagram shows a uniform lamina \(ABCD\) with \(AB = 0.75 \text{ m}\), \(AD = 0.6 \text{ m}\) and \(BC = 0.9 \text{ m}\). Angle \(BAD =\) angle \(ABC = 90°\).

1. Show that the distance of the centre of mass of the lamina from \(AB\) is \(0.38 \text{ m}\), and find the distance of the centre of mass from \(BC\). [5]

The lamina is freely suspended at \(B\) and hangs in equilibrium.

1. Find the angle between \(BC\) and the vertical. [2]

\includegraphics{figure_1} \(ABCD\) is a uniform lamina with \(AB = 1.8\) m, \(AD = DC = 0.9\) m, and \(AD\) perpendicular to \(AB\) and \(DC\) (see diagram).

1. Find the distance of the centre of mass of the lamina from \(AB\) and the distance from \(AD\). [4]

The lamina is freely suspended at \(A\) and hangs in equilibrium.

1. Calculate the angle between \(AB\) and the vertical. [2]

A uniform circular disc has centre \(O\) and radius \(1.2\text{ m}\). The centre of the disc is at the origin of \(x\)- and \(y\)-axes. Two circular holes with centres at \(A\) and \(B\) are made in the disc (see diagram). The point \(A\) is on the negative \(x\)-axis with \(OA = 0.5\text{ m}\). The point \(B\) is on the negative \(y\)-axis with \(OB = 0.7\text{ m}\). The hole with centre \(A\) has radius \(0.3\text{ m}\) and the hole with centre \(B\) has radius \(0.4\text{ m}\). Find the distance of the centre of mass of the object from

1. the \(x\)-axis, [4]
2. the \(y\)-axis. [3]

The object can rotate freely in a vertical plane about a horizontal axis through \(O\).

1. Calculate the angle which \(OA\) makes with the vertical when the object rests in equilibrium. [2]

\includegraphics{figure_6} A uniform circular disc has centre \(O\) and radius \(1.2\,\text{m}\). The centre of the disc is at the origin of \(x\)- and \(y\)-axes. Two circular holes with centres at \(A\) and \(B\) are made in the disc (see diagram). The point \(A\) is on the negative \(x\)-axis with \(OA = 0.5\,\text{m}\). The point \(B\) is on the negative \(y\)-axis with \(OB = 0.7\,\text{m}\). The hole with centre \(A\) has radius \(0.3\,\text{m}\) and the hole with centre \(B\) has radius \(0.4\,\text{m}\). Find the distance of the centre of mass of the object from

1. the \(x\)-axis, [4]
2. the \(y\)-axis. [3]

The object can rotate freely in a vertical plane about a horizontal axis through \(O\).

1. Calculate the angle which \(OA\) makes with the vertical when the object rests in equilibrium. [2]

\includegraphics{figure_2} A uniform wire is bent to form an object which has a semicircular arc with diameter \(AB\) of length 1.2 m, with a smaller semicircular arc with diameter \(BC\) of length 0.6 m. The end \(C\) of the smaller arc is at the centre of the larger arc (see diagram). The two semicircular arcs of the wire are in the same plane.

1. Show that the distance of the centre of mass of the object from the line \(ACB\) is 0.191 m, correct to 3 significant figures. [3]

The object is freely suspended at \(A\) and hangs in equilibrium.

1. Find the angle between \(ACB\) and the vertical. [4]

\includegraphics{figure_4} The diagram shows the cross-section \(ABCD\) through the centre of mass of a uniform solid prism. \(AB = 0.9\) m, \(BC = 2a\) m, \(AD = a\) m and angle \(ABC =\) angle \(BAD = 90°\).

1. Calculate the distance of the centre of mass of the prism from \(AD\). [2]
2. Express the distance of the centre of mass of the prism from \(AB\) in terms of \(a\). [2]

The prism has weight 18 N and rests in equilibrium on a rough horizontal surface, with \(AD\) in contact with the surface. A horizontal force of magnitude 6 N is applied to the prism. This force acts through the centre of mass in the direction \(BC\).

1. Given that the prism is on the point of toppling, calculate \(a\). [3]

\includegraphics{figure_2} A uniform object is made by attaching the base of a solid hemisphere to the base of a solid cone so that the object has an axis of symmetry. The base of the cone has radius \(0.3\text{ m}\), and the hemisphere has radius \(0.2\text{ m}\). The object is placed on a horizontal plane with a point \(A\) on the curved surface of the hemisphere and a point \(B\) on the circumference of the cone in contact with the plane (see diagram).

1. Given that the object is on the point of toppling about \(B\), find the distance of the centre of mass of the object from the base of the cone. [3]
2. Given instead that the object is on the point of toppling about \(A\), calculate the height of the cone. [3]

[The volume of a cone is \(\frac{1}{3}\pi r^2 h\). The volume of a hemisphere is \(\frac{2}{3}\pi r^3\).]

\includegraphics{figure_6} Fig. 1 shows the cross-section \(ABCDE\) through the centre of mass \(G\) of a uniform prism. The cross-section consists of a rectangle \(ABCF\) from which a triangle \(DEF\) has been removed; \(AB = 0.6\text{ m}\), \(BC = 0.7\text{ m}\) and \(DF = EF = 0.3\text{ m}\).

1. Show that the distance of \(G\) from \(BC\) is \(0.276\text{ m}\), and find the distance of \(G\) from \(AB\). [5]
2. The prism is placed with \(CD\) on a rough horizontal surface. A force of magnitude \(2\text{ N}\) acting in the plane of the cross-section is applied to the prism. The line of action of the force passes through \(G\) and is perpendicular to \(DE\) (see Fig. 2). The prism is on the point of toppling about the edge through \(D\). Calculate the weight of the prism. [3]

\includegraphics{figure_4} The diagram shows the cross-section \(ABCD\) of a uniform solid object which is formed by removing a cone with cross-section \(DCE\) from the top of a larger cone with cross-section \(ABE\). The perpendicular distance between \(AB\) and \(DC\) is \(h\), the diameter \(AB\) is \(6r\) and the diameter \(DC\) is \(2r\).

1. Find an expression, in terms of \(h\), for the distance of the centre of mass of the solid object from \(AB\). [4]

The object is freely suspended from the point \(B\) and hangs in equilibrium. The angle between \(AB\) and the downward vertical through \(B\) is \(\theta\).

1. Given that \(h = \frac{13}{4}r\), find the value of \(\tan\theta\). [2]

\includegraphics{figure_4} An object is formed by removing a solid cylinder, of height \(ka\) and radius \(\frac{1}{2}a\), from a uniform solid hemisphere of radius \(a\). The axes of symmetry of the hemisphere and the cylinder coincide and one circular face of the cylinder coincides with the plane face of the hemisphere. \(AB\) is a diameter of the circular face of the hemisphere (see diagram).

1. Show that the distance of the centre of mass of the object from \(AB\) is \(\frac{3a(2-k^2)}{2(8-3k)}\). [4] When the object is freely suspended from the point \(A\), the line \(AB\) makes an angle \(\theta\) with the downward vertical, where \(\tan\theta = \frac{7}{18}\).
2. Find the possible values of \(k\). [3]

\includegraphics{figure_1} Figure 1 shows a template made by removing a square \(WXYZ\) from a uniform triangular lamina \(ABC\). The lamina is isosceles with \(CA = CB\) and \(AB = 12a\). The midpoint of \(AB\) is \(N\) and \(NC = 8a\). The centre \(O\) of the square lies on \(NC\) and \(ON = 2a\). The sides \(WX\) and \(ZY\) are parallel to \(AB\) and \(WZ = 2a\). The centre of mass of the template is at \(G\).

1. Show that \(NG = \frac{8}{7}a\). [7]

The template has mass \(M\). A small metal stud of mass \(kM\) is attached to the template at \(C\). The centre of mass of the combined template and stud lies on \(YZ\). By modelling the stud as a particle,

1. calculate the value of \(k\). [4]

\includegraphics{figure_1} A uniform lamina \(L\) is formed by taking a uniform square sheet of material \(ABCD\), of side 10 cm, and removing the semi-circle with diameter \(AB\) from the square, as shown in Fig. 2.

1. Find, in cm to 2 decimal places, the distance of the centre of mass of the lamina \(L\) from the mid-point of \(AB\). [7]

[The centre of mass of a uniform semi-circular lamina, radius \(a\), is at a distance \(\frac{4a}{3π}\) from the centre of the bounding diameter.] The lamina is freely suspended from \(D\) and hangs at rest.

1. Find, in degrees to one decimal place, the angle between \(CD\) and the vertical. [4]

\includegraphics{figure_1} Figure 1 shows a template made by removing a square \(WXYZ\) from a uniform triangular lamina \(ABC\). The lamina is isosceles with \(CA = CB\) and \(AB = 12a\). The mid-point of \(AB\) is \(N\) and \(NC = 8a\). The centre \(O\) of the square lies on \(NC\) and \(ON = 2a\). The sides \(WX\) and \(ZY\) are parallel to \(AB\) and \(WZ = 2a\). The centre of mass of the template is at \(G\).

1. Show that \(NG = \frac{30}{11}a\). [7]

The template has mass \(M\). A small metal stud of mass \(kM\) is attached to the template at \(C\). The centre of mass of the combined template and stud lies on \(YZ\). By modelling the stud as a particle,

1. calculate the value of \(k\). [4]